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A Quantum-safe Key Exchange Scheme using Mihailova Subgroups in Braid groups

Hanling Lin, Yu Han

TL;DR

This work addresses the quantum-era security challenges of braid-based cryptography by replacing the conventional conjugacy-based hardness with the algorithmically unsolvable membership problem for Mihailova subgroups within braid groups $B_n$ for $n\ge 6$. It introduces a modified AAG key exchange where private keys are drawn from Mihailova subgroups of $B_n/\langle \Delta^2\rangle$, leveraging the central element $\Delta^2$ to tie conjugation to an unsolvable membership problem. The authors argue that any attempt to recover the shared key reduces to solving the unsolvable membership problem in these Mihailova subgroups, thereby resisting both classical and quantum attacks. They provide explicit subgroup constructions, parameter regimes, and a formal security rationale, presenting braid-based cryptography as a concrete quantum-safe candidate. This constitutes a theoretically grounded advancement toward post-quantum cryptography in non-abelian group settings.

Abstract

In this paper,we propose a modified Anshel-Anshel-Goldfeld(AAG) key exchange scheme. The hardness assumption underlying this modified construction is based on the membership problem for Mihailova subgroups of the braid group, a problem that is algorithmically unsolvable. According to the security analysis, we show that the proposed scheme is resistant to all known attacks, including quantum computational attacks.

A Quantum-safe Key Exchange Scheme using Mihailova Subgroups in Braid groups

TL;DR

This work addresses the quantum-era security challenges of braid-based cryptography by replacing the conventional conjugacy-based hardness with the algorithmically unsolvable membership problem for Mihailova subgroups within braid groups for . It introduces a modified AAG key exchange where private keys are drawn from Mihailova subgroups of , leveraging the central element to tie conjugation to an unsolvable membership problem. The authors argue that any attempt to recover the shared key reduces to solving the unsolvable membership problem in these Mihailova subgroups, thereby resisting both classical and quantum attacks. They provide explicit subgroup constructions, parameter regimes, and a formal security rationale, presenting braid-based cryptography as a concrete quantum-safe candidate. This constitutes a theoretically grounded advancement toward post-quantum cryptography in non-abelian group settings.

Abstract

In this paper,we propose a modified Anshel-Anshel-Goldfeld(AAG) key exchange scheme. The hardness assumption underlying this modified construction is based on the membership problem for Mihailova subgroups of the braid group, a problem that is algorithmically unsolvable. According to the security analysis, we show that the proposed scheme is resistant to all known attacks, including quantum computational attacks.
Paper Structure (9 sections, 3 theorems, 16 equations)

This paper contains 9 sections, 3 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Braid groups and Mihailova subgroups
  3. Review of Anshell-Anshell-Goldfeld scheme and Attacks
  4. Anshell-Anshell-Goldfeld scheme
  5. Decision problems and Normal forms
  6. Attacks against Anshell-Anshell-Goldfeld scheme
  7. The reformed Anshell-Anshell-Goldfeld scheme
  8. Parameters and Security Analysis
  9. Conclusion

Key Result

Theorem 2.1

(Mihailova, Mi) The membership problem for $M(H)$ in $F_{k}\times F_{k}$ is solvable if and only if the word problem for $H$ is solvable.

Theorems & Definitions (3)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 4.1